Modular Form Data by Ken-ichi SHIOTA
Summary
I have calculated the follwing data on the spaces M
2(Γ
0(q))
of the elliptic modular forms of weight 2,
for all prime levels 11 <= q < 10000:
- The (q,∞)-quaternion algebra D over Q
- A maximal order O in D
- The class number H and the type number T of D
- H is equal to the dimension of M2(Γ0(q)).
- T is equal to the dimension of the minus-eigenspace M2-(Γ0(q)) of the Atkin-Lehner involution.
- The representatives { Aj } j = 1,...,H of the left O-ideal classes
- The maximal orders Oj = (Aj)-1Aj in D
- The representatives { Aij } i = 1,...,H of the left Oj-ideal classes
- Aij is obtained by Aij = ( a constant multiple of ) (Aj)-1Ai.
- The group order ej of the unit group of Oj
- The theta series θij associated to the norm form of Aij
- The definition of the theta series is
    
θij(z) = (1/ej) Σ x in Aij exp(2πiz N(x)/N(Aij))
where N denotes the reduced norm on D.
- We have ej θij = ei θji, hence we have to calculate θij only for i >= j.
- The Brandt matrices B(n) ( n = 0,1,2,... )
- The definition of the Brandt matrices is
    
Σ n = 0,1,2,... B(n) exp(2πinz) = ( θij(z) ) i,j = 1,...,H
- For n >= 1, B(n) is an integral representation matrix of the Hecke operator T(n) on M2(Γ0(q)).
- The characteristic polynomial of B(n) ( = that of T(n) ) and its factorization over Z
Further, I am calculating the follwing data:
- The common eigenvectors of B(n)'s, and the eigenvalues of B(n0) for a selected n0
- Each eigenvector corresponds to the Eisenstein series E or a newform f in the space of the elliptic cusp forms S2(Γ0(q)).
- Each eigenvalue is equal to a Hecke eigenvalue, hence to a Fourier coefficient of E or f.
Reference
- [HPS] Hiroaki Hijikata, Arnold K. Pizer and Thomas R. Shemanske, The basis problem for modular forms on Γ0(N),
Mem. Amer. Math. Soc. 82 (1989), no. 418.
- [Pi80]
A. Pizer, An algorithm for computing modular forms on Γ0(N), Journal of Algebra 64,(1980), 340-390.
- [Shio91]
Ken-ichi Shiota, On theta series and the splitting of S2(Γ0(q)), J. Math. Kyoto Univ., 31 (1991.12), 909-930.
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